Dependent Types and Dynamics of Natural Language

Dependent Types Dynamics in NLS DTS Conclusion
Dependent Types
and
Dynamics of Natural Language
Daisuke Bekki
Ochanomizu University, Faculty of Core Science
A talk at Inui Lab, Tohoku University
27 July, 2022.
1 / 61

Dependent Type Semantics (DTS) (Bekki 2014;
Bekki and Mineshima 2017; Bekki 2021)
I A framework of natural language semantics
I Unified approach to general inferences and
anaphora/presupposition resolution in terms of proof
construction (cf. Krahmer and Piwek (1999))
Main features:
1. Proof-theoretic semantics:
From truth-conditions (denotations, models) to
proof-conditions (proofs, contexts)
2. Underspecification Semantics: A proof-theoretic alternative
to Dynamic Semantics (DRT, DPL, etc.)
3. Compositional semantics: Syntax-semantics interface via
categorial grammars (e.g. CCG, Hybrid-TLCG)
4. Computational semantics: Implementation, Applications to
Natural Language Processing
2 / 61

Dependent Types
3 / 61

Per Martin-Löf
Martin-Löf (1984) “Intuitionistic type theory”
4 / 61

What are Π-types
Π-type is a type of fibred functions.
Simple function space Fibred function space
5 / 61

What are Σ-types
Σ-type is a type of fibred products.
Simple product space Fibred product space
6 / 61

Notations
DTS notation Standard notation x 6∈ fv(B) x ∈ fv(B)
(x : A) → B (Πx : A)B A → B (∀x : A)B
(x : A) × B
or

x : A
B
(Σx : A)B A ∧ B (∃x : A)B
Scope of the variable in Π-types: (x : A) → B
Scope of the variable in Σ-types:

x : A
B

7 / 61

Π-type F/I/E rules
A : type
x : A
i
.
.
.
.
B : type
(x : A) → B : type
(ΠF),i
A : type
x : A
i
.
.
.
.
M : B
λx.M : (x : A) → B
(ΠI ),i
M : (x : A) → B N : A
MN : B[N/x]
(ΠE)
8 / 61

Σ-type F/I/E rules
A : type
x : A
i
.
.
.
.
B : type
(x : A) × B : type
(ΣF),i
M : A N : B[M/x]
(M, N) : (x : A) × B
(ΣI )
M : (x : A) × B
π1(M) : A
(ΣE)
M : (x : A) × B
π2(M) : B[π1(M)/x]
(ΣE)
9 / 61

Rules of DTS
Rules from Martin-Löf Type Theory
I Axioms and Structural rules
I Π-type (Dependent function type) [F/I/E]
I Σ-type (Dependent product type) [F/I/E]
I Intensional equality type [F/I/E]
I Disjoint union type [F/I/E]
I Enumeration type [F/I/E]
I Natural number type [F/I/E]
New rule in DTS
I @ (the ‘asperand’ operator)
I Anaphora and presupposition triggers
(linguistically speaking)
I Open proofs (logically speaking)
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Conjunction, Implication, and Negation
Definition

A
B

def
≡ (x : A) × B where x /
∈ fv(B).
A → B
def
≡ (x : A) → B where x /
∈ fv(B).
¬A
def
≡ (x : A) → ⊥
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Dynamics in Natural Language
Semantics
12 / 61

A theory of anaphora
I Anaphora representable by a constant symbol:
I Deictic use:
(1) (Pointing at John)
He was born in Detroit.
bornIn( j , d)
I Coreference:
(2) John loves a girl who hates him .
∃x(girl(x) ∧ love( j , x) ∧ hate(x, j ))
I Anaphora representable by a variable
I Bound variable anaphora:
(3) Every boy loves his father.
∀x (boy(x) → love(x, fatherOf( x )))
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A theory of anaphora
I Anaphora not representable by FoL:
I E-type anaphora:
(4) A man entered into the park. He whistled.
I Donkey anaphora:
(5) Every farmer who owns a donkey beats it .
(6) If a farmer owns a donkey , he beats it .
I Anaphora not representable by FoL nor dynamic semantics:
I Syllogistic anaphora:
(7) Every girl received a present . Some girl opened it .
I Disjunctive antecedent:
(8) If Mary sees a horse or a pony , she waves to it .
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E-type anaphora: Evans (1980)
(9) [A man]1 entered. He1 whistled.
The first-order SR (10) represents the truth condition of (9), thus
is a candidate of the SR of (9).
(10) ∃x(man(x) ∧ enter(x) ∧ whistle(x))
But the syntactic structure of the SR (10) does not correspond to
that of (9), where consists of two independent sentences. The
sentential boundary of (9) prefers the first-order
representation (11).
(11) ∃x(man(x) ∧ enter(x)) ∧ whistle( x )
However, the truth condition of (11) is different from that of the
mini-discourse (9) since the variable x in whistle(x) is not bound
by ∃.
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Donkey anaphora: Geach (1962)
For the donkey sentences (12), a first-order formula (13), whose
truth condition is the same as those of (12), is a candidate of its
SR.
(12) a. Every farmer who owns [a donkey]1 beats it1.
b. If [a farmer]1 owns [a donkey]2 , he1 beats it2.
(13) ∀x(farmer(x) → ∀y (donkey(y) ∧ own(x, y) →
beat(x, y)))
But the translation from the sentence (12) to (13) is not
straightforward since i) the indefinite noun phrase a donkey is
translated into a universal quantifier in (13) instead of an
existential quantifier, and ii) the syntactic structure of (13) does
not corresponds to that of (12).
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Donkey anaphora: Geach (1962)
(12) a. Every farmer who owns [a donkey]1 beats it1 .
b. If [a farmer]1 owns [a donkey]2, he1 beats it2 .
The syntactic parallel of (12) is, rather, the SR (14), in which the
indefinite noun phrase is translated into an existential
quantification.
(14) ∀x(farmer(x) ∧ ∃y(donkey(y) ∧ own(x, y)) →
beat(x, y ))
However, (14) does not represent the truth condition of (12)
correctly since the variable y in beat(x, y) fails to be bound by ∃.
Therefore, neither (13) nor (14) qualifies as the SR of (12).
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E-type anaphora: Ranta (1994)
(9) A man entered. He whistled.







u :



x : entity

man(x)
enter(x)




whistle( π1(u) )







Note:

x : A
B

is a type for pairs of A and B[x].
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Donkey anaphora: Sundholm (1986)
(12a) Every farmer who owns a donkey beats it .







u :







x : entity





farmer(x)



v :

y : entity
donkey(y)
#
own(x, π1v)






















→ beat(π1u, π1π1π2π2u
Note: (x : A) → B is a type for functions from A to B[x].
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From TTG to DTS: Compositionality
Q: How could one get to these (dependently-typed)
representations from arbitrary sentences?
A: By lexicalization.
Q: How could we lexicalize context-dependent words like
pronouns?







u :



x : entity

man(x)
enter(x)




whistle( π1(u) )







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Dependent Type Semantics (DTS)
21 / 61

From TTG to DTS: Compositionality
Q: How could one get to these (dependently-typed)
representations from arbitrary sentences?
A: By lexicalization.
Q: How could we lexicalize context-dependent words like
pronouns?
A: By using underspecified terms.
Q: But how could we retrieve a context for an
underspecified term?
A: By type checking.
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Underspecified terms
DTS = DTT + underspecified terms @A
Definition (@-rule)
A : typej A true
@iA : A
where j ∈ N
(@)
I @-rule states that the well-formedness of @A requires:
I A is a well-formed type.
I the inhabitance of A (namely, A is a presuppositional content)
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On parsing
We assume that an SR of a sentence is obtained by parsing and
semantic composition (assumed by one’s syntactic theory).
A sentence
⇓
Parsing
⇓
An underspecified SRs in DTS
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Lexical items in CCG-style
PF CCG categories Semantic representations in DTS
if S/S/S λp.λq. (u : p) → q
everynom S/(SNP)/N λn.λp.

u :

x : entity
nx

→ p(π1(u))
everyacc T (T /NP)/N λn.λp.λ~
x.

v :

y : entity
ny

→ p(π1(v))~
x
anom , somenom S/(SNP)/N λn.λp.

 u :

x : entity
nx

p(π1(u))


aacc, someacc T (T /NP)/N λn.λp.λ~
x.

 v :

y : entity
ny

p(π1(v))~
x


farmer N farmer
donkey N donkey
who NN/(SNP ) λp.λn.λx.

nx
px

whom NN/(S/NP ) λp.λn.λx.

nx
px

owns SNP/NP own
beats SNP/NP beat
he NP π1 @

x : entity
male(x)
!
it NP π1 @

x : entity
¬human(x)
!
the NP/N λn.π1 @

x : entity
nx
!
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Lexical items in CCG-style (anaphoric expressions)
PF CCG categories Semantic representations in DTS
he NP π1 @

x : entity
male(x)
!
it NP π1 @

x : entity
¬human(x)
!
the NP/N λn.π1 @

x : entity
nx
!
26 / 61

E-type anaphora: Parsing
A
S/(SNP)/N
: λn.λp.

 u :

x : entity
nx

p(π1u)


man
N
: λx.man(x)
S/(SNP)
: λp.

 u :

x : entity
man(x)

p(π1u)



entered
SNP
: λx.enter(x)
S
:

 u :

x : entity
man(x)

enter(π1u)



27 / 61

E-type anaphora: Parsing
He
NP
: π1 @

x : entity
male(x)

whistled
SNP
: λx.whistle(x)
S
: whistle π1 @

x : entity
male(x)
!

28 / 61

Progressive conjunction: Ranta (1994)
Definition (Progressive conjunction)
M; N
def
≡

u : M
N

where u /
∈ fv(N)
(9) [A man]1 entered. He 1 whistled.


x : entity

man(x)
enter(x)


 ; whistle(π1 @

x : entity
male(x)

)
=







u :


x : entity

man(x)
enter(x)



whistle(π1 @

x : entity
male(x)

)







29 / 61

E-type anahora: Type checking
.
.
.

 u :

x : entity
man(x)

enter(π1u)

 : type
whistle
: entity
→ type
(CON )
.
.
.

x : entity
male(x)

: type
v :

 u :

x : entity
man(x)

enter(π1u)


1
.
.
.
.

x : entity
male(x)

true
@

x : entity
male(x)

:

x : entity
male(x)
(@)
π1 @

x : entity
male(x)

: entity
(ΣE)
whistle π1 @

x : entity
male(x)
!
: type
(→E)







v :

 u :

x : entity
man(x)

enter(π1u)



x : entity
!







: type
(ΣF),1
30 / 61

E-type anaphora: Proof search
v :

 u :

x : entity
man(x)

enter(π1u)


1
π1v :

x : entity
man(x)
(ΣE)
π1π1v : entity
(ΣE)
v :

 u :

x : entity
man(x)

enter(π1u)


1
π1v :

x : entity
man(x)
(ΣE)
m :

u :

x : entity
man(x)

→ male(π1
m(π1v) : male(π1π1v)
(π1π1v, m(π1v)) :

x : entity
male(x)
(ΣE)
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E-type anaphora: @-elimination
.
.
.
.

 u :

x : entity
man(x)

enter(π1u)

 : type
whistle : entity → type
v :

 u :

x : entity
man(x)

enter(π1u)


1
.
.
.
.
(π1π1v, m(π1v)) :

x : entity
male(x)

π1π1v : entity
(ΣE)
whistle( π1π1v ) : type
(ΠE)




v :

 u :

x : entity
man(x)

enter(π1u)


whistle( π1π1v )



 : type
(ΣF),1
32 / 61

@-elimination rules
Definition (@-elimination rules (excerpt))
u
w
vA : s
DM
M : A0
@iA : A
(@)
}

~ =
DM
M : A0
u
w
w
v
DA
A : s1
x : A0
DB
B : s2
(x : A) → B : s2
(ΠF)
}

~ =
JDAK
A0 : s1
x : A0
JDBK
B0 : s2
(x : A0) → B0 : s2
(ΠF)
u
w
w
v
DA
A : s1
x : A0
DM
M : B
λx.M : (x : A) → B
(ΠI )
}

~ =
JDAK
A0 : s1
x : A0
JDM K
M0 : B0
λx.M0 : (x : A0) → B0
(ΠI )
u
w
v
DM
M : (x : A) → B
DN
N : A
MN : B0
(ΠE)
}

~ =
JDM K
M0 : (x : A0) → B0
JDN K
N0 : A0
M0N0 : B00
where B0[N0/x] β B00
(ΠE)
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The model of language understanding
A sentence . . . A sentence
⇓ ⇓
Parsing . . . Parsing
⇓ ⇓
An underspecified SRs in DTS . . . An underspecified SRs in DTS
⇓ ⇓
Discoruse relation (e.g. Progressive conjunction)
⇓
An underspecified discourse representation in DTS
⇓
Type checking + Proof search in DTS
⇓
A proof diagram of the well-formedness of an SR in DTS
⇓
@-elimination
⇓
A proof diagram of the well-formedness of an SR in DTT
⇓
Inference in DTT
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Concluding Remarks
35 / 61

Compositional Theory of Anaphora
I DTS provides a unified analysis for (general) inferences and
anaphora resolusion mechanisms (at least) for:
I Deictic use and coreference
I Bound variable anaphora (BVA)
I E-type anaphora
I Donkey anaphora
I Bridging anaphora
I Syllogistic anaphora
I Disjunctive antecedents
36 / 61

Compositional Theory of Anaphora
I The background theory for DTS is an extention of DTT with
underspecified terms and the @-rule .
I Lexical items of anaphoric expressions and presupposition
triggers are represented by using underspecified terms.
I Context retrieval in DTS reduces to type checking .
I Anaphora resolution and presupposition binding in DTS
reduces to proof search .
I @-elimination translates a proof diagram of DTS into a proof
diagram of DTT, by which an SR in DTT is obtained with all
anaphora resolved.
37 / 61

Natural language semantics via dependent types:
Classics
I Donkey anaphora: Sundholm (1986)
I Translation from DRS to dependent type representations: Ahn
and Kolb (1990)
I Summation: Fox (1994a,b)
I Ranta’s TTG (Relative and Implicational Donkey Sentences,
Branching Quantifiers, Intensionality, Tense): Ranta (1994)
I Translation from Montague Grammar to dependent type
representations: Dávila-Pérez (1995)
I Presupposition Binding and Accommodation, Bridging:
Krahmer and Piwek (1999), Piwek and Krahmer (2000)
38 / 61

Natural language semantics via dependent types:
Recent frameworks
I Type Theory with Record (TTR): Cooper (2005)
I Modern Type Theory: Luo (1997, 1999, 2010, 2012), Asher
and Luo (2012), Chatzikyriakidis (2014)
I Semantics with Dependent Types: Grudzinska and
Zawadowski (2014; 2017)
I Dependent Type Semantics (DTS): Bekki (2014), Bekki
and Mineshima (2017)
I (Dynamic Categorial Grammar: Martin and Pollard (2014))
39 / 61

Semantic Analyses by DTS
I Generalized Quantifiers: Tanaka (2014)
I Honorification: Watanabe et al. (2014)
I Conventional Implicature: Bekki and McCready (2015)
I Factive Presuppositions: Tanaka et al. (2015)(2017)
I Dependent Plural Anaphora: Tanaka et al. (2017)
I Paycheck sentences: Tanaka et al. (2018) in NLCS2018 (July
7)
I Coercion and Metaphor: Kinoshita et al. (2018)
I Questions in Dependent Type Semantics: Watanabe et al.
(NLCS’19)
I Comparision with DRT: Yana et al. (2019) in JoLLI
40 / 61

Semantic Analyses by DTS
I Development of an automated theorem prover for the
fragment of DTS: Daido and Bekki (2020) in LENLS17
I A Proof-theoretic Analysis of Weak Crossover: Bekki (2021)
in LENLS18
I The proviso problem from a proof-theoretic perspective: Yana
et al. (2021) in LACL2021
I Integrating Deep Neural Network with Dependent Type
Semantics: Bekki et al. (2021) in LACompLing2021
I Learning Knowledge with Neural DTS: Bekki et al. (2022) in
NALOMA’22
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Thank you!
42 / 61

Typing Rules @-Elim. Rules References
Typing Rules in DTS
43 / 61

Axioms and Structural Rules
A : type
x : A
(VAR)
c : A
(CON )
where σ ` c : A.
M : A N : B
M : A
(WK)
M : A
M : B
(CONV )
where A =β B.
44 / 61

Π-type F/I/E rules
A : type
x : A
i
.
.
.
.
B : type
(x : A) → B : type
(ΠF),i
A : type
x : A
i
.
.
.
.
M : B
λx.M : (x : A) → B
(ΠI ),i
M : (x : A) → B N : A
MN : B[N/x]
(ΠE)
45 / 61

Σ-type F/I/E rules
A : type
x : A
i
.
.
.
.
B : type
(x : A) × B : type
(ΣF),i
M : A N : B[M/x]
(M, N) : (x : A) × B
(ΣI )
M : (x : A) × B
π1(M) : A
(ΣE)
M : (x : A) × B
π2(M) : B[π1(M)/x]
(ΣE)
46 / 61

Disjoint Union Type F/I/E rules
A : type B : type
A ] B : type
(]F)
M : A
ι1(M) : A ] B
(]I )
N : B
ι2(N) : A ] B
(]I )
L : A ] B P : (A ] B) → type M : (x : A) → P (ι1(x)) N : (x
unpackP
L (M, N) : P (L)
47 / 61

Natural Number Type F/I/E rules
N : type
(NF)
0 : N
(NI )
n : N
s(n) : N
(NI )
n : N P : N → type e : P (0) f : (k : N) → P (k) → P (s(k))
natrecP
n (e, f) : P (n)
48 / 61

Enumeration Type F/I/E rules
{a1, . . . , an} : type
({}F)
ai : {a1, . . . , an}
({}I )
M : {a1, . . . , an} P : {a1, . . . , an} → type N1 : P (a1) . . . Nn : P (an)
caseP
M (N1, . . . , Nn) : P (M)
({}E)
49 / 61

Intensional Equality Type F/I/E rules
A : type M : A N : A
M =A N : type
(=F)
A : type M : A
reflA(M) : M =A M
(=I )
E : M =A N P : (x : A) → (y : A) → (x =A y) → type R : (x : A) → P xx(reflA(x))
idpeelP
E (R) : P MNE
(=
50 / 61

@-rule
A : typej A true
@iA : A
where j ∈ N
(@)
51 / 61

@-Elimination Rules in DTS
52 / 61

Axioms
u
w
vA : s
DM
M : A0
@iA : A
(@)
}

~ =
DM
M : A0
s
c : A
(CON )
{
= c : A
(CON )
r
x : A
z
= x : A
s
type : kind
(typeF)
{
= type : kind
(typeF)
53 / 61

Π-type
u
w
w
v
DA
A : s1
x : A0
DB
B : s2
(x : A) → B : s2
(ΠF)
}

~ =
JDAK
A0 : s1
x : A0
JDBK
B0 : s2
(x : A0) → B0 : s2
(ΠF)
u
w
w
v
DA
A : s1
x : A0
DM
M : B
λx.M : (x : A) → B
(ΠI )
}

~ =
JDAK
A0 : s1
x : A0
JDM K
M0 : B0
λx.M0 : (x : A0) → B0
(ΠI )
u
w
v
DM
M : (x : A) → B
DN
N : A
MN : B0
(ΠE)
}

~ =
JDM K
M0 : (x : A0) → B0
JDN K
N0 : A0
M0N0 : B00
where B0[N0/x] β B00
(ΠE)
54 / 61

Σ-type
u
w
w
w
v
DA
A : s1
x : A0
DB
B : s2

x : A
B

: s2
(ΣF)
}

~
=
JDAK
A0 : s1
x : A0
JDBK
B0 : s2

x : A0
B0

: s2
(ΣF)
u
w
w
v
DM
M : A
DN
N : B0
(M, N) :

x : A
B
(ΣI )
}

~ =
JDM K
M0 : A0
JDN K
N0 : B000
(M0, N0) :

x : A0
B00

where B00[M0/x] β B000
(ΣI )
u
w
w
w
v
DM
M :

x : A
B

π1(M) : A
(ΣE)
}

~
=
JDM K
M0 :

x : A0
B0

π1(M0) : A0
(ΣE)
u
w
w
w
v
DM
M :

x : A
B

π2(M) : B0
(ΣE)
}

~
=
JDM K
M0 :

x : A0
B00

π2(M0) : B000
(ΣE)
55 / 61

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Reference III
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61 / 61

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